Lesson This nice problem teaches to distinguish permutations from combinations

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This nice problem teaches to distinguish permutations from combinations

Problem 1

A scholarship committee has $3000 to award this year and has 12 qualified candidates.
Thom thinks that individual awards of $1500, $1000, and $500 would be appropriate,
while Peter thinks that 3 awards of $1000 each would seem logical.
Count the number of possible outcomes for each plan.

Solution

If we consider Thom's plan, then the number of all possible distinguishable triples
is the product 12*11*10 = 1320 of three integer numbers, starting from 12 in the descending order.


It is so, because the order inside each such triple does matter:

for example, triple (Alex, Berta, Carl) is different from triple (Berta, Alex, Carl),
because the participants obtain different awards.



If we consider Peter's plan, then the number of all possible distinguishable triples
is the number of combinations   =  =  = 220  of three participants
taken at a time from 12 participants..


It is so, because the order inside each such triple does NOT matter:

for example, triple (Alex, Berta, Carl) is not distinguishable from triple (Berta, Alex, Carl),
because the participants obtain the same awards.

Thus the problem is solved completely.

It is a nice problem: it teaches a reader to distinguish between permutations and combinations, depending on context.

My other additional lessons on Combinatorics problems in this site are

    - Upper league combinatorics problem
    - Upper level problems on a party of people sitting at a round table
    - Upper level combinatorics problem on subsets of a finite set
    - A confusing combinatorics problem on repeating digits in numbers
    - Upper level combinatorics problems on Inclusion-Exclusion principle
    - Upper level combinatorics problem on finding the number of arrangements along a straight line
    - OVERVIEW of additional combinatorics problems

Use this file/link ALGEBRA-II - YOUR ONLINE TEXTBOOK to navigate over all topics and lessons of the online textbook ALGEBRA-II.

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